83 research outputs found
A time dependent Stokes interface problem: well-posedness and space-time finite element discretization
In this paper a time dependent Stokes problem that is motivated by a standard
sharp interface model for the fluid dynamics of two-phase flows is studied.
This Stokes interface problem has discontinuous density and viscosity
coefficients and a pressure solution that is discontinuous across an evolving
interface. This strongly simplified two-phase Stokes equation is considered to
be a good model problem for the development and analysis of finite element
discretization methods for two-phase flow problems. In view of the unfitted
finite element methods that are often used for two-phase flow simulations, we
are particularly interested in a well-posed variational formulation of this
Stokes interface problem in a Euclidean setting. Such well-posed weak
formulations, which are not known in the literature, are the main results of
this paper. Different variants are considered, namely one with suitable spaces
of divergence free functions, a discrete-in-time version of it, and variants in
which the divergence free constraint in the solution space is treated by a
pressure Lagrange multiplier. The discrete-in-time variational formulation
involving the pressure variable for the divergence free constraint is a natural
starting point for a space-time finite element discretization. Such a method is
introduced and results of numerical experiments with this method are presented
A partition of unity approach to fluid mechanics and fluid-structure interaction
For problems involving large deformations of thin structures, simulating
fluid-structure interaction (FSI) remains challenging largely due to the need
to balance computational feasibility, efficiency, and solution accuracy.
Overlapping domain techniques have been introduced as a way to combine the
fluid-solid mesh conformity, seen in moving-mesh methods, without the need for
mesh smoothing or re-meshing, which is a core characteristic of fixed mesh
approaches. In this work, we introduce a novel overlapping domain method based
on a partition of unity approach. Unified function spaces are defined as a
weighted sum of fields given on two overlapping meshes. The method is shown to
achieve optimal convergence rates and to be stable for steady-state Stokes,
Navier-Stokes, and ALE Navier-Stokes problems. Finally, we present results for
FSI in the case of a 2D mock aortic valve simulation. These initial results
point to the potential applicability of the method to a wide range of FSI
applications, enabling boundary layer refinement and large deformations without
the need for re-meshing or user-defined stabilization.Comment: 34 pages, 15 figur
Nondifferentiable energy minimization for cohesive fracture in a discontinuous Galerkin finite element framework
Until recently, most works on the computational modelling of fracture relied on a Newtonian mechanics approach, i.e., momentum balance equations describing the motion of the body along with fracture criteria describing the evolution of fractures. Robustness issues associated with this approach have been identified in the previous literature, several of which, as this thesis shows, due to the discontinuous dependence of stress field on the deformation field at the time of insertion of displacement discontinuities. Lack of continuity limits applicability of the models and undermines reliability of the numerical solutions. In particular, solutions often show non-convergent behaviour with time step refinement and exhibit nonphysical velocity fields and crack activation patterns. In addition, implicit time-stepping schemes, which are favoured in quasi-static and low-velocity problems, are challenging in such models. This is not a coincidence but a manifestation of algorithmic pitfalls of such methods.
Continuity of stresses is in general hard to achieve in a computational model that employs a crack initiation criterion. Energy (variational) approaches to fracture have gained increased popularity in recent years. An energy approach has been shown to avoid introduction of a crack initiation criterion. The central idea of this model is the minimization of a mechanical energy functional, whose term representing the energy due to the cracks is a nondifferentiable function of the
interface openings at zero opening displacement. A consequence
of this formulation is that crack initiation happens automatically as a by-product of energy minimization. This avoids the complexities arising from the introduction of an extrinsic activation criterion but entails minimization of a nondifferentiable potential.
The aim of this research is to develop robust and efficient computational algorithms for numerical implementation of the energy approach to cohesive fracture. Two computational algorithms have been proposed in a discontinuous Galerkin finite element framework, including a continuation algorithm which entails successive smooth
approximations of the nondifferentiable functional and a block coordinate descent algorithm which uses generalized differential calculus for the treatment of nondifferentiability. These methods allow for a seamless transition from the uncracked to the cracked state, making possible the use of iterative solvers with implicit time-stepping, and completely sidestepping robustness issues of previous computational frameworks.
A critical component of this work is validation of the robustness of the proposed numerical methods. Various numerical simulations are presented including time step and mesh size convergence studies and qualitative and quantitative comparison of simulations with experimental observations and theoretical findings. In addition, an energy-based hydro-mechanical model and computational algorithm is presented for hydraulic fracturing in impermeable media, which shows the crucial importance of continuity in multi-physics modelling. A search algorithm is developed on the basis of graph theory to identify the set of fluid-pressurized cracks among cracks in naturally fractured domains
A fourth-order unfitted characteristic finite element method for solving the advection-diffusion equation on time-varying domains
We propose a fourth-order unfitted characteristic finite element method to
solve the advection-diffusion equation on time-varying domains. Based on a
characteristic-Galerkin formulation, our method combines the cubic MARS method
for interface tracking, the fourth-order backward differentiation formula for
temporal integration, and an unfitted finite element method for spatial
discretization. Our convergence analysis includes errors of discretely
representing the moving boundary, tracing boundary markers, and the spatial
discretization and the temporal integration of the governing equation.
Numerical experiments are performed on a rotating domain and a severely
deformed domain to verify our theoretical results and to demonstrate the
optimal convergence of the proposed method
3D Nitsche-XFEM method for fluid-structure interaction with immersed thin-walled solids
This paper extends the unfitted Nitsche-XFEM method of [Comput. Methods Appl. Mech. Engrg., 301, 300-335, 2016] to three-dimensional fluid-structure interaction problems with immersed thin-walled elastic solids. The fluid and solid domains are discretized with unfitted unstructured meshes. Discrete weak and strong discontinuities are allowed in the fluid and the coupling is enforced consistently via a fluid-sided Nitsche's type mortaring with suitable stabilization for robustness. Integration over cut-elements is handled via an efficient and robust intersection and subtesselation algorithm. The method includes a new approach for the treatment of partially intersected fluid domains. Several numerical examples are presented and discussed, which illustrate the capabilities of the proposed method
An Eulerian Finite Element Method for PDEs in time-dependent domains
The paper introduces a new finite element numerical method for the solution
of partial differential equations on evolving domains. The approach uses a
completely Eulerian description of the domain motion. The physical domain is
embedded in a triangulated computational domain and can overlap the
time-independent background mesh in an arbitrary way. The numerical method is
based on finite difference discretizations of time derivatives and a standard
geometrically unfitted finite element method with an additional stabilization
term in the spatial domain. The performance and analysis of the method rely on
the fundamental extension result in Sobolev spaces for functions defined on
bounded domains. This paper includes a complete stability and error analysis,
which accounts for discretization errors resulting from finite difference and
finite element approximations as well as for geometric errors coming from a
possible approximate recovery of the physical domain. Several numerical
examples illustrate the theory and demonstrate the practical efficiency of the
method.Comment: 27 pages, 3 figures, 8 table
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